The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c} a\text{ mod q}\\ (a,q)=1 \end{array}}\left|\pi(y;q,a)-\frac{y}{\phi(q)\log y}\right|\ll \frac{x}{(\log x)^A}.$$
Here it was asked what happens when we restrict $q$ so that they are all divisible by some smaller integer $k.$
I was wondering if we can obtain a stronger bound if we restrict $q$ such that $q\in S$ for some small set $S$ (say $|S|=O(\log \log x)$). Normally, applying Siegel-Walfisz Theorem to such small sets $S$ consisting of numbers at most $(\log x)^c$ for some $c>0$ would give bounds similar to Bombieri-Vinogradov Theorem but one wonders if Bombieri-Vinogradov Theorem applied to scarce sets gives better error terms?
